SubGroup

Given a group G under an operation *, we say that some subset H of G is a SubGroup if H is a group under the restriction of * thereto. This occurs if and only if H is closed to products and inverses.

The subgroups of any given group form a lattice under inclusion. There is a minimal subgroup, the trivial group {e}, and a maximal subgroup, the group itself. The minimal subgroup containing some set S is denoted <S> and is said to be generated by S. Groups generated by a single element are called cyclic and are isomorphic to either the IntegerNumbers or some ModularArithmetic.

A group HomoMorphism f:G->K sends subgroups of G to subgroups of K. Moreover, the preimage of any subgroup of K is a subgroup of G. Of particular importance is the preimage of {e}, called the kernel of the homomorphism. Any kernel H always satisfies the property g*H=H*g for all g in G. Such subgroups are called normal. There is a homomorphism f with any given normal subgroup H as its kernel, and moreover the homomorphic images under all such f are isomorphic to the QuotientGroup? G/H.